L(s) = 1 | + (−0.669 + 0.743i)2-s + (0.5 − 0.866i)3-s + (−0.104 − 0.994i)4-s + (0.309 + 0.951i)6-s + (−0.978 + 0.207i)7-s + (0.809 + 0.587i)8-s + (−0.5 − 0.866i)9-s + (−0.669 + 0.743i)11-s + (−0.913 − 0.406i)12-s + (−0.104 − 0.994i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)16-s + (−0.809 + 0.587i)17-s + (0.978 + 0.207i)18-s + (0.104 − 0.994i)19-s + ⋯ |
L(s) = 1 | + (−0.669 + 0.743i)2-s + (0.5 − 0.866i)3-s + (−0.104 − 0.994i)4-s + (0.309 + 0.951i)6-s + (−0.978 + 0.207i)7-s + (0.809 + 0.587i)8-s + (−0.5 − 0.866i)9-s + (−0.669 + 0.743i)11-s + (−0.913 − 0.406i)12-s + (−0.104 − 0.994i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)16-s + (−0.809 + 0.587i)17-s + (0.978 + 0.207i)18-s + (0.104 − 0.994i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1011183656 + 0.1304306205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1011183656 + 0.1304306205i\) |
\(L(1)\) |
\(\approx\) |
\(0.6110480480 - 0.05473373891i\) |
\(L(1)\) |
\(\approx\) |
\(0.6110480480 - 0.05473373891i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (-0.669 + 0.743i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.978 + 0.207i)T \) |
| 11 | \( 1 + (-0.669 + 0.743i)T \) |
| 13 | \( 1 + (-0.104 - 0.994i)T \) |
| 17 | \( 1 + (-0.809 + 0.587i)T \) |
| 19 | \( 1 + (0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.309 - 0.951i)T \) |
| 29 | \( 1 + (-0.104 + 0.994i)T \) |
| 31 | \( 1 + (-0.913 - 0.406i)T \) |
| 37 | \( 1 + (0.669 - 0.743i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.309 + 0.951i)T \) |
| 47 | \( 1 + (0.809 + 0.587i)T \) |
| 53 | \( 1 + (-0.913 + 0.406i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + (-0.913 - 0.406i)T \) |
| 71 | \( 1 + (0.104 - 0.994i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.913 + 0.406i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.42718834852190167634984584231, −16.74923956080073582353260440811, −16.259220200614121199540494246548, −15.858617451102099042326627522921, −15.058989715375397983224651584716, −14.013539349817755907608587677560, −13.47421536289912357149874364648, −13.105658197285218870235700893806, −11.99036730672377388945246482432, −11.42601045560927769326603381547, −10.798822712282615681718591106949, −10.00682253308443439401746917648, −9.71760205541910160327609407300, −8.92638857979370482101496660473, −8.521485821756622825561764161008, −7.58770181716181649926516560439, −7.00889682889123847655085843703, −5.96736665635676258784999273345, −5.10886761242529200621505339270, −4.09534923913858596972700954506, −3.74778422374751552880296658005, −2.89909327455072380528418596879, −2.41550982832608964186935686884, −1.40899001029278295829570688909, −0.06775772384260676585999278544,
0.68018821102700377152045789533, 1.82818201587168984494806216421, 2.50124948173917045673908239219, 3.17652082442346942338987833478, 4.35235634245939718778994597480, 5.25549032210717048284077877147, 5.97492291325839454264208251654, 6.6537708246455340230958277607, 7.1575472307291298928400423897, 7.797363061783105996945478860785, 8.46836840288329615013206899117, 9.21829357332434128197123521606, 9.55063014796686607726277398220, 10.63696997931690377493612179263, 10.94833861253170188709335125556, 12.28067100633679483725954120731, 12.882786849995378486647757115650, 13.16019766545397572405771046681, 14.031192500196237334637465317345, 14.94590864453300193409616810586, 15.17684362060259559444722164160, 15.892860912665112498971226412079, 16.61119861304730267041175814706, 17.4830345424163948704179840692, 17.91021289647825942295884115644